Asymptotics of blowup solutions for the aggregation equation

Yanghong Huang; Andrea Bertozzi

doi:10.3934/dcdsb.2012.17.1309

Article Contents

2012, Volume 17, Issue 4: 1309-1331. Doi: 10.3934/dcdsb.2012.17.1309 open access

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Asymptotics of blowup solutions for the aggregation equation

Yanghong Huang^1, and
Andrea Bertozzi^2,

1.
Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC V5A 1S6
2.
520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095

Received: December 31, 2010

Revised: August 31, 2011

Published: May 2012

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Abstract

We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation $ u_t = \nabla\cdot(u\nabla K*u) $ in $\mathbb{R}^n$, for homogeneous potentials $K(x) = |x|^\gamma$, $\gamma>0$. For $\gamma>2$, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing $\delta$-ring. We develop an asymptotic theory for the approach to this singular solution. For $\gamma < 2$, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all $\gamma$ in this range, including additional asymptotic behaviors in the limits $\gamma \to 0^+$ and $\gamma\to 2^-$.

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Mathematics Subject Classification: Primary: 35B40, 35B44; Secondary: 92D50.

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References

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